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<a href="http://fmota91.blogspot.com/">A Different Kind of Uselessness</a>
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<h2 class="date-header">Wednesday, September 19, 2007</h2>
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<a href="http://fmota91.blogspot.com/2007/09/small-shots-of-lambda-calculus-part-i.html">Small Shots of Lambda Calculus, Part I</a>
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Lambda calculus was invented by Alonzo Church.<br><br>Lambda calculus revolves around the application of anonymous functions -- lambda expressions.<br><br>[For the sake of clarity, I'll be using the syntax <code>(a -&gt; b)</code> over <code>(lambda a. b)</code>.]<br><br>It's best to explain by example.<br><br><code>(a -&gt; b) x</code> becomes <code>b</code><br><code>(a -&gt; a) x</code> becomes <code>x</code><br><code>(a b -&gt; b a) x y</code> becomes <code>y x</code><br><code>(a b -&gt; b a) (a -&gt; b) (a -&gt; a)</code> becomes <code>(a -&gt; a) (a -&gt; b)</code> becomes <code>(a -&gt; b)</code><br><br>Everything is a function in this extremely minimal lambda calculus. I'll introduce you to some of the most important ones:<br><br><code>I = (x -&gt; x)</code><br><br><code>I</code> doesn't do much -- it just returns its argument.<br><br><code>true = (a b -&gt; a)</code><br><code>false = (a b -&gt; b)</code><br><br><code>true</code> returns its first argument while <code>false</code> returns its second argument. They're the archetypal Boolean values of lambda calculus.<br><br><code>0 = (f x -&gt; x)<br>1 = (f x -&gt; f x)<br>2 = (f x -&gt; f (f x))<br>3 = (f x -&gt; f (f (f x)))<br>...</code><br><br>These
are called the Church numerals, and they're used to apply a function to
an argument a variable number of times. Note the similarity (in fact,
equivalence) between 0 and false.<br><br><code>succ = (n f x -&gt; f (n f x))</code><br><br>We can use succ to go from one Church numeral to the next. It is now trivial to do arithmetic on Church numerals.<br><br><code>add = (a b -&gt; a succ b)</code><br><code>mul = (a b -&gt; a (add b) 0)</code><br><br>G'night. Tomorrow: More lambda calculus.
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